[Paper Review] Adaptive Multiple Importance Sampling

Jean‐Marie Cornuet, Jean‐Michel Marin+2 more|arXiv (Cornell University)|Jul 7, 2009

Bayesian Methods and Mixture Models28 references7 citations

TL;DR

This paper proposes Adaptive Multiple Importance Sampling (AMIS), an algorithm that recursively reuses all past and current simulations by recomputing their importance weights using a deterministic mixture of all proposal densities. AMIS achieves substantial variance reduction and efficiency gains—demonstrated by a 6x speedup in a population genetics case study—by fully recycling past samples and adaptively refining proposal distributions at each iteration.

ABSTRACT

The Adaptive Multiple Importance Sampling (AMIS) algorithm is aimed at an optimal recycling of past simulations in an iterated importance sampling scheme. The difference with earlier adaptive importance sampling implementations like Population Monte Carlo is that the importance weights of all simulated values, past as well as present, are recomputed at each iteration, following the technique of the deterministic multiple mixture estimator of Owen and Zhou (2000). Although the convergence properties of the algorithm cannot be fully investigated, we demonstrate through a challenging banana shape target distribution and a population genetics example that the improvement brought by this technique is substantial.

Motivation & Objective

• Address the inefficiency of standard importance sampling in high-dimensional problems due to weight degeneracy and poor proposal selection.
• Overcome limitations of existing adaptive importance sampling methods like Population Monte Carlo (PMC), which do not fully recycle past samples or optimize over all historical data.
• Develop a method that adaptively improves proposal distributions using all past weighted samples, enhancing sampling efficiency and reducing computational cost.
• Demonstrate the practical superiority of AMIS over MCMC and standard IS in complex, high-dimensional models such as those in population genetics.
• Provide a framework for integrating importance sampling with sequential learning that maximizes reuse of simulations without requiring resampling or reweighting from scratch.

Proposed method

• At each iteration $ t $, all past and current samples $ \mathbf{y}_i^l $ for $ l = 0, \dots, t $, are reweighted using a deterministic mixture of all proposal densities $ q_l $, forming a global denominator: $ \frac{1}{\sum_{j=0}^T N_j} \sum_{l=0}^T N_l q_l(\mathbf{y}_i^t) $.
• The importance weight for each sample becomes $ \omega_i^t = \pi(\mathbf{y}_i^t) / \left( \frac{1}{\sum_{j=0}^T N_j} \sum_{l=0}^T N_l q_l(\mathbf{y}_i^t) \right) $, ensuring unbiased estimation of expectations under the target $ \pi $.
• The proposal distribution $ Q_{t+1} $ at each step is constructed adaptively from the full history of weighted samples, allowing the algorithm to learn from all previous simulations.
• The method avoids resampling and maintains all particles throughout iterations, enabling full recycling of simulations and minimizing redundant likelihood evaluations.
• The algorithm is applied in a sequential manner: at each iteration, weights are updated and the next proposal is refined based on the entire weighted sample set.
• AMIS is implemented in a population genetics model using IS-based likelihood estimation, where genealogies are simulated and weighted using importance sampling with adaptive proposals.

Experimental results

Research questions

• RQ1 Can a multiple importance sampling scheme that reuses all past simulations with dynamically updated weights achieve better efficiency than standard or PMC-based adaptive IS?
• RQ2 How does the deterministic mixture weighting strategy affect the variance and convergence of the estimator in high-dimensional or challenging target distributions?
• RQ3 To what extent does adaptive proposal refinement using all historical samples improve sampling efficiency in complex models like those in population genetics?
• RQ4 Can AMIS significantly reduce computational cost compared to MCMC while maintaining or improving estimation accuracy and repeatability?
• RQ5 What is the impact of full simulation recycling on the stability and reliability of posterior inference in models with expensive likelihood evaluations?

Key findings

• AMIS achieved a 6x speedup over MCMC in a population genetics case study, reducing runtime from ~2 hours to ~20 minutes for the same inference task.
• The IS-AMIS method produced posterior distributions that were more consistent across repeated runs than MCMC, indicating improved repeatability and stability.
• The posterior estimates from AMIS and MCMC were highly similar, validating the accuracy of the AMIS approximation despite its non-Markovian, non-rejection-based structure.
• The algorithm’s ability to recycle all past simulations reduced the effective number of likelihood evaluations, significantly lowering computational cost in models with expensive likelihood computation.
• The deterministic mixture weighting scheme preserved unbiasedness of the estimator while stabilizing the variance of the importance weights across iterations.
• No prior calibration of the initial importance function was needed, as the prior distribution was already sufficiently concentrated to explore the relevant parameter space effectively.